# Numerical method for wave equation

Hi, I am trying to plot a function subjected to a nonlinear wave equation. One of the method I found for solving the nonlinear schrodinger equation is the split step fourier method. However I noticed that this method only works for a specific form of PDE where the equation has an analytic solution for both the linear and nonlinear part. So how do I solve numerically any arbitrary PDE? Specifically, my PDE is

$\partial_zE(r_⊥,z,\tau)-∇^2_⊥\int^{\tau}_{-\infty}d\tau 'E(r_⊥,z,\tau ')=\int^{\tau}_{-\infty}d\tau '\omega^2(r_⊥,z,\tau ')-\frac{\partial_{\tau}n(r_⊥,z,\tau)}{E(r_⊥,z,\tau)}$

and I want to solve for the evolution of E. Any help would be greatly appreciated. Thanks!

From my experience with PDE, you only need boundary conditions when solving for specific solutions. I am looking for a way to solve the PDE numerically. Would it be possible to give me the general method? For example you can $\bf{explain}$ the split step Fourier method without knowing the boundary conditions?